Consider a BDQE:
$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$
where $ A,B,C,D,E,F \in \mathbb Z$
Is there a method to determine (prove/disprove) if integer solution(s) to this equation exist(s) without actually calculating it/them?
Asked
Active
Viewed 50 times
0
-
This problem is equivalent to finding iteger points on conics. – Mar 24 '17 at 09:01
-
And...? :) A little more? – plktrautman Mar 24 '17 at 09:03
-
I guess no. If the equation leads to a pell-like equation , I do not think that there is a way to show that a solution exists without calculating at least one. – Peter Mar 24 '17 at 09:46
-
That's a pity... – plktrautman Mar 24 '17 at 10:17
-
Can be reduced to this equation and then use the formula. The problem is reduced to finding the equivalent of the equation Pell. https://artofproblemsolving.com/community/c3046h1048219 – individ Mar 24 '17 at 12:37
1 Answers
0
Ok, the general problem of solvability of Diophantine equations (not only BDQE, all of them) is the content of 10th Hilbert's Problem which was solved with a negative answer (no general algorithm exists). Still, I thought there could be an exception for specific subclasses of DE like BDQE, but again this is not true.
plktrautman
- 95
-
Just a moment. There IS a method to find all solutions of the equation you have given, but I think there is no way to find out whether solutions exist WITHOUT concrete calculation. The equation you mentioned is however decideable! – Peter Mar 24 '17 at 11:32
-
In the sense of a solution, Hilbert wanted, this equation is easy enough that the answer whether an algorithm exists would be "yes". – Peter Mar 24 '17 at 11:36
-
I know there is a method to find them. But you cannot determine the existence of the solutions unless you enter into calculating at least one of them. – plktrautman Mar 24 '17 at 12:11